We present a new approach to symmetry breaking for pairs of real forms of $(\mathit{GL}(n, \mathbb{C}), \mathit{GL}(n-1, \mathbb{C}))$. While translation functors are a useful tool for studying a family of representations of a single reductive group $G$, when applied to a pair of groups $G \supset G'$, translation functors can significantly alter the nature of symmetry breaking between the representations of $G$ and $G'$, even within the same Weyl chamber of the direct product group $G \times G'$. We introduce the concept of “fences for the interlacing pattern”, which provides a refinement of the usual notion of “walls for Weyl chambers”. We then present a theorem that states that multiplicity is constant unless these “fences” are crossed. This general theorem is illustrated with examples of both tempered and non-tempered representations. Additionally, we provide a new non-vanishing theorem of period integrals for pairs of reductive symmetric spaces, which is further strengthened through this approach.
[ arXiv | preprint version(pdf) ]
© Toshiyuki Kobayashi